Consider the matrix A =
| 7 16 8|
|-1 0 -1|
|-2 -10 -3|
Show that A is diagonalizable. Find an invertible matrix P and diagonal matrix D and use the obtained result in order to calculate A^2 and A^3
Determinant equation I suppose.
D is the diagonal matrix, like identity matrix but constructed from the eigenvalues. The order of the eigenvalues must match the columns in P precisely.
The Attempt at a Solution
Alright well I've done a huge chunk of this problem already. I found the eigenvalues to be 2, 3, and -1. Then I find my eigenvectors, and I come up with 2 per eigenvalue.
For λ = 2 I get [-2 1 0] and [-1 0 1]
For λ = 3 I get [-3 1 0] and [-1 0 1]
For λ = 1 I get [1 1 0] and [0 1 1]
So now I have to construct P from these. This is where I'm confused. Which eigenvectors from which eigenvalues do I use? I've tried several combinations to make AP = PD and I just can't do it. Everything comes out wrong. I suppose there's just something I'm not understanding. Please help.